I am hopelessly confused on a homework assignment.
The problem says " (a) Find the series radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?"
You are rewriting (4x+1)n to something like (x+...)n.
You're not doing it right though.
You are dividing by 4 inside the brackets, and then multiply by 4 outside the brackets. What you should do is multiply by 4n outside the brackets.
(4x+1)n = 4n(x + 1/4)n
However I don't think this will lead to a different answer...
You most likely want the root test to find whether those series converge (absolutely) or diverge. As for finding the radius of convergence, see the bit related to the root test here.
Determining whether those series converge absolutely or conditionally for certain x's, though, might be trickier. I'd have to look at it more, though I suspect it turns into reducing some of the problems to the (alternating) harmonic series for some specific value of x. I bet most of those can actually be done by inspection too.
In the above links, ignore the "lim sup" and just think "lim", since the "sup" part is only a technicality you won't have to bother with unless you take an Analysis course. If these problems don't make sense in light of the above links, though, you might want to talk to your instructor because these concepts would be difficult to pick up on from examples alone.
The time you enjoy wasting is not wasted time. Bertrand Russell
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